Various forms of the Fourier series description for periodic signals are based on alternate ways of writing a cosine signal. For a signal
x(t) = a cos(w t + q )
with amplitude a > 0, frequency w
> 0, and phase angle q
, three additional expressions for x(t) are (with 
)
Trigonometric: 
Complex exponential: 
Phasor real part: 
Equivalence of these expressions can be verified by using the Euler formula and standard trigonometric identities.
We will explore properties of Fourier series using the phasor representation. A phasor
can be viewed as a vector at the origin of the complex plane having length a and, at any time t, angle (w t + q ). Thus the vector rotates counterclockwise with time, since w > 0, and the projection on the real axis is described by
For graphical representation, projection on the real (horizontal) axis is inconvenient, and therefore we rotate phasors by p /2 radians and project on the vertical axis. This makes use of the mathematical relationship
Click on the complex plane below to define a vector of length a and initial angle ( at t = 0) of (q + p /2) radians. The corresponding phasor and cosine waveform will be shown with a convenient frequency w .
In a similar fashion, a sum of cosine waveforms can be represented as the real part of a phasor sum:
The sum of two vectors in the complex plane can be found by placing the vectors "head to tail." Click twice on the complex plane below to define a vector sum, and the corresponding two-cosine waveform with radian frequencies of w and 2w will be displayed.
A Fourier series is a (possibly infinite) sum of cosine waveforms with the property that every frequency is a positive integer multiple of a fundamental frequency, 
. We write this as
where 
and each 
.
Terminology for the various terms is listed below:
harmonic term: 
The terms in such an expression are said to be harmonically related. This feature implies periodicity of x(t), since each term in the sum repeats, at least once, in any time interval of length 
.
Simple examples of Fourier series are
w(t) = 1 + 4 cos(3t) + 2 cos(6t + p /4)
which has fundamental frequency 3, and
y(t) = 1 + 4 cos(3t) + 2 cos(4t + p /4)
which has fundamental frequency 1. However,
z(t) = 1 + 4 cos(t) + 2 cos(p t)
is not a Fourier series since the frequencies 1 and p cannot be written as integer multiples of a single fundamental frequency.
The phasor representation for a Fourier series has the form
Of course in practice the infinite sum is truncated to a finite number of terms. Explore below the truncated Fourier series for a triangular wave, a square wave, and a periodic train of impulses
These Fourier series are given by
For a periodic signal that is a continuous function of time, such as the triangle wave, the Fourier series coefficients diminish at least as fast as 
. That is, for some positive constant M,
You can observe in the applet above that it takes very few terms to achieve a good approximation to the triangle wave.
For a discontinuous function, such as the square wave, the coefficients diminish as 1/k. This raises mathematical issues of convergence, particularly at points of discontinuity, and gives rise to the Gibbs effect. The Gibbs effect is the overshoot phenomenon exhibited by the truncated Fourier series at points of discontinuity. This behavior is apparent with the square wave in the applet. Notice that as more terms are added to the Fourier series, the overshoot near the discontinuity decreases only slightly in amplitude, though it decreases significantly in duration.
A more extreme case is the impulse train, where the Fourier series coefficients remain constant and the mathematical nature of convergence of the series is far from apparent. From the truncated series in the applet, it is clear that the Gibbs effect is present, and also that there is a type of convergence.
There are several methods for modifying the coefficients in a truncated Fourier series to reduce or eliminate Gibbs effect. Essentially these are different ways of weighting (decreasing) the coefficient values, and are referred to as windowing methods. One method is the Fejer Window, which is based on Fejer summation of series. In this method, if N harmonics are included in the truncated Fourier series, then the amplitude of the kth harmonic is multiplied by (N - k)/N. Thus including the first 5 harmonics (some have zero amplitude) for the square wave and impulse train yields the expressions
A second method is the Hamming Window, where the kth harmonic in an N harmonic series is multiplied by
For the square wave and impulse train examples, this yields the expressions
Notice that both windows apply a unity multiplier to the constant term (k = 0), and that as N increases, the kth-term multipliers tend toward unity for both windows. That is, the limit function of the series is unchanged. The success of these two approaches in eliminating the Gibbs effect can be explored in the applet below.
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Applet by Hsi Chen Lee
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